Mathematical Sciences: Fourier Analysis and Partial Differential Equations

在这项数学研究的过程中，将涉及三个主要领域：几何、分析和经济建模。黎曼几何的Yamabe问题是问，如果新的度量是通过乘以正函数从旧的度量得到的，则具有给定的黎曼度量的流形是否与具有常数量曲率的流形共形等价。我们将对柯西-黎曼结构的类似问题进行研究。在这里，人们试图选择具有恒定的伪厄米标量曲率的李维形式。这相当于求解流形上的显式偏微分方程式。关于紧致严格伪凸可定向流形的结果在奇数维上得到了推广。其余案件的工作将继续进行。还将努力推广函数的幂积分与函数在热运算符下的映象之间的基本(P，Q1)估计。过去20年的结果虽然很尖锐，但都建立在两个指数是二元性的假设之上。这些结果都符合索博列夫不等式的一般定义。对这类比较的新兴趣与偏微分方程解的唯一性性质的工作有关。除了寻求新的Soblev不等式外，还将努力将它们的适用性扩展到更一般的微分算子。最近对经济学的兴趣源于弗罗贝尼乌斯关于微分方程的一个结果，在现代术语中，该方程表明一种形式何时具有积分因子。在微观经济学中，人们考虑一种形式的差异，即收入和需求差异、收入和价格函数(乘以价格差异)之间的差异。这些形式区分切线向量是否指向改进的方向。弗罗贝尼乌斯结果是效用函数存在的陈述--它通常被认为是一个基本公理。有证据表明，这一公理对消费者群体来说是失败的。本课题将集中于刻画可以期望近似可积性的条件。

Three major areas will be addressed during the course of this mathematical research; geometry, analysis and economic modeling. The Yamabe problem of Riemannian geometry asks if a manifold with a given Riemannian metric is conformally equivalent to another with constant scalar curvature if the new metric is obtained from the old by multiplication by a positive function. Work will be done on the analogous problem for Cauchy-Riemann structures. Here one seeks to choose a Levi form with constant pseudohermitian scalar curvature. This amounts to solving an explicit partial differential equation on the manifold. Results on compact strictly pseudoconvex orientable manifolds have been extensive for odd dimensions. Work will proceed on the remaining cases. Efforts will also be made to extend fundamental (P, q1) - estimates between power integrals of a function and the image of the function under the heat operator. Results over the past two decades, while sharp, have rested on the assumption that the two exponents are in duality. Such results all fall within the general definition of Sobolev inequalities. The renewed interest in such comparisons is related to work on uniqueness properties of solutions of partial differential equations. In addition to seeking new Sobolev inequalities work will also be done in expanding their applicability to more general differential operators. Recent interest in economics derives from a result of Frobenius about differential equations which, in modern terminology, states when a one-form has an integrating factor. In microeconomics one considers one-forms of differences between differentials of income and demand functions of income and prices (multiplied by differentials of prices). These forms distinguish whether or not a tangent vector is pointing in the direction of improvement. The Frobenius result is the statement that a utility function exists - it is generally regarded as a basic axiom. There is evidence to suggest that the axiom fails for collections of consumers. The present project will focus on characterizing conditions when approximate integrability can be expected.